Ordered Ranked Posets, Representations of Integers and Inequalities from Extremal Poset Problems

Some New Results on Macaulay Posets

“…This poset also was studied by Sali [50,52] with respect to Sperner and intersecting properties. For n ≤ 3 the poset M n is Macaulay, but not for n ≥ 4 in contradistinction to a conjecture in [28].…”

“…For example, if P is a poset whose Hasse diagram is isomorphic to K p,p for p ≥ 2 (i.e. we have a special case of a so-called complete poset [28]) then P × P is not Macaulay in contradistinction to a conjecture in [28]. Indeed, if m ≤ p, then a set of m elements of P 2 1 has minimal shadow iff these elements agree in some entry whose rank in P is 0.…”

“…Daykin [28] introduced the V -order , an extension of the VIP-order for SO(n) with n ≥ 2. He conjectured that this order is a Macaulay order for SO(n).…”